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We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain ${\varOmega }\subset \mathbb {R}^{N} (N \geq 2)$ . We prove the existence of entropy solutions avoiding sign condition and coercivity on the lower order terms.

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a p-Laplacian and Dirichlet conditions $$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u(t))+|u(t)|^{p-2}u(t)= \lambda f(t,u(t)), &{}t\ne t_j,\ \ t\in (0,T),\\ \varDelta (D^{\alpha -1}_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u))(t_j)=I_j(u(t_j)),\\ u(0)=u(T)=0, \end{array}\right. } \end{aligned}$$ where $$\alpha \in (\frac{1}{p}, 1]$$ and $$p > 1$$ . Their approach is based on variational methods. The main result is illustrated with an example.

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

A linear wave equation with acoustic boundary conditions (ABC) on a portion of the boundary and Dirichlet conditions on the rest of the boundary is considered. The (ABC) contain a memory damping with respect to the normal displacement of the boundary point. In this paper, we establish polynomial energy decay rates for the wave equation by using resolvent estimates.

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions. Before to do this, we construct a discrete operator whose fixed points coincide with the solutions of the problem we are looking for. Moreover, we introduce a strong positiveness condition on the related Green’s function that allows us to construct suitable cones where to apply adequate fixed point theorems. Once we have the general existence result, we deduce, as a particular case, the existence of solutions of a second order difference equation with nonlocal perturbed Dirichlet conditions.

We consider the transformation of the initial data space for the Schrodinger equation. The transformation is generated by nonlinear Schrodinger operator on the segment [−π, π] satisfying the homogeneous Dirichlet conditions on the boundary of the segment. The potential here has the type $\xi(u)=\left(1+|u|^{2}\right)^{\frac{p}{2}} u$ , where u is an unknown function, p ≥ 0. The Schrodinger operator defined on the Sobolev space $$H_0^2([-\pi, \pi])$$ generates a vector field $${\rm{v}}:H_0^2([-\pi, \pi])\rightarrow{H}\equiv{L_2}(-\pi, \pi)$$ . First, we study the phenomenon of global existence of a solution of the Cauchy problem for p ∈ [0, 4) and, second, the phenomenon of rise of a solution gradient blow up during a finite time for p ∈ [4, +∞). In second case we study qualitative properties of a solution when it approaches to the boundary of its interval of existence. Moreover, we define a solution extension through the moment of a gradient blow up using both the one-parameter family of multifunctions and the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states.

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0

We consider a Steklov-type problem for the Laplace operator in a half-strip containing a small hole with the Dirichlet conditions on the lateral boundaries and the boundary of the hole and the Steklov spectral condition on the base of the half-strip. We prove that eigenvalues of this problem vanish as the small parameter (the “diameter” of the hole) tends to zero.

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. Intuitively, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary. Our approach is based on real analysis techniques.

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

This article is devoted to deduce the expression and the main properties of the Green’s function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In particular, it is proved that such function has constant sign on their set of definition and, moreover, it satisfies some additional strong sign conditions that are fundamental to define suitable cones, where to ensure the existence of solutions of nonlinear problems.

We study the Dirichlet problem of the n-dimensional complex Monge—Ampere equation det(u ij ) = F/|z|2α, where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program.

We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described.

A new graph-theoretical approach is presented for heat diffusion analysis given an arbitrary geometry and spatially varying parameters. A weighted directed graph with positive communication or conduction weights is proposed for heat transfer analysis, where the communication weights of the graph are determined based on the positions of the nodes and the spatially varying thermal and material properties of the domain. Transient Neumann and Dirichlet conditions are considered at the boundary nodes of the graph. Then, the temperature at each interior node is updated based on the temperatures at its in-neighbor nodes, where the in-neighbor vertices of interior nodes and the conduction weights are assigned based on the conduction graph. A set of coupled first-order ordinary differential equations determines the transient temperatures at the interior nodes for a prescribed boundary condition. The proposed method can be applied to both steady-state and transient heat diffusion analysis.

Under study are the two classes of elliptic spectral problems with homogeneous Dirichlet conditions and discontinuous nonlinearities (the parameter occurs in the nonlinearity multiplicatively). In the former case the nonlinearity is nonnegative and vanishes for the values of the phase variable not exceeding some positive number c; it has linear growth at infinity in the phase variable u and the only discontinuity at u = c. We prove that for every spectral parameter greater than the minimal eigenvalue of the differential part of the equation with the homogeneous Dirichlet condition, the corresponding boundary value problem has a nontrivial strong solution. The corresponding free boundary in this case is of zero measure. A lower estimate for the spectral parameter is established as well. In the latter case the differential part of the equation is formally selfadjoint and the nonlinearity has sublinear growth at infinity. Some upper estimate for the spectral parameter is given in this case.

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

We present two primal methods to weakly discretize (linear) Dirichlet and (nonlinear) Signorini boundary conditions in elliptic model problems. Both methods support polyhedral meshes with nonmatching interfaces and are based on a combination of the hybrid high-order (HHO) method and Nitsche’s method. Since HHO methods involve both cell unknowns and face unknowns, this leads to different formulations of Nitsche’s consistency and penalty terms, either using the trace of the cell unknowns (cell version) or using directly the face unknowns (face version). The face version uses equal-order polynomials for cell and face unknowns, whereas the cell version uses cell unknowns of one order higher than the face unknowns. For Dirichlet conditions, optimal error estimates are established for both versions. For Signorini conditions, optimal error estimates are proven only for the cell version. Numerical experiments confirm the theoretical results and also reveal optimal convergence for the face version applied to Signorini conditions.

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points., Comment: 37 pages

The article investigates the growth of multiple Dirichlet series. The lower order and the linear order of n-tuple Dirichlet series in ℂn are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained.

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schr\"odinger equations with time independent coefficients., Comment: 36 pages

In this paper we consider the behavior of funcions defined by Dirichlet series with multiplicative coefficients and with bounded summatory function when approaching the imaginary axis. We show that the points of the imaginary axis are also the points of continuity in a broad sense of functions defined by Dirichlet series with multiplicative coefficients which are determined by nonprincipal generalized characters. This result is particularly interesting in its connection with a solution of Chudakov hyphotesis, which states that any finite-valued numerical character, which does not vanish on all prime numbers and has bounded summatory function, is a Dirichlet character. The proof of the main result in this paper is based on the method of reduction to power series, basic principles of which were developed by prof. Kuznetsov in the early 1980s. Ths method establishes a connection between analytical properties of Dirichlet series and boundary properties of the corresponding power series (i.e. a power series with the same coefficients as the Dirichlet series). This allows to obtain new results both for the Dirichlet series and for the power series. In our case this method allowed us to prove the main result using the properties of the power series with multiplicative coefficients determined by the nonprincipal generalized characters, which also were obtained in this work.

In this paper we study the problem of analytical behavior of Dirichlet series with a bounded summatory function on its axis of convergence, σ = 0. This problem was also considered in the authors’ earlier works in case of Dirichlet series with coefficients determined by finite-valued numerical characters, which, in turn, was connected with a solution for a well-known Chudakov hypothesis. The Chudakov hypothesis suggests that generalized characters, which do not vanish on almost all prime numbers p and asymptotic behavior of whose summatory functions is linear, are Dirichlet characters. This hypothesis was proposed in 1950 and was not completely proven until now. A partial proof based on the behavior of a corresponding Dirichlet series when it approaches to the imaginary axis was obtained in one of authors’ works. There are reasons to anticipate that this approach may eventually lead to a full proof of the Chudakov hypothesis. In our case this problem is particularly interesting in connection with finding analytical conditions of almost periodic behavior of a bounded number sequence, different from those obtained before by various authors, for example, by Szego. Our study is based on a so called method of reduction to power series. This method was developed by Prof. V. N. Kuznetsov in the 1980s and it consists in studying the relation between the analytical properties of Dirichlet series and the boundary behavior of the corresponding (i.e. with the same coefficients) power series. In our case this method of reduction to power series allowed us to show that such Dirichlet series are continuous in the wide sense on the entire imaginary axis. Moreover, this method also helped to construct a sequence of Dirichlet polynomials which converge to a function determined by a Dirichlet series in any rectangle inside the critical strip.

We present the spectral properties of an even-order differential operator whose domain is described by periodic and antiperiodic boundary conditions or the Dirichlet conditions. We derive an asymptotic formula for the eigenvalues, estimates for the deviations of spectral projections, and estimates for the equiconvergence rate of spectral decompositions. Our asymptotic formulas for eigenvalues refine well-known ones.

Gakhov class G is formed by the holomorphic and locally univalent functions in the unit disk with no more than unique critical point of the conformal radius. Let D be the classical Dirichlet space, and let P: f ↦ F = f″/f′. We prove that the radius of the maximal ball in P(G)∩D with the center at F = 0 is equal to 2.

It is well known that the potential q of the Sturm–Liouville operator L y = −yʺ + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum $$\left\{ {{\lambda _k}} \right\}_1^\infty $$ and the normalizing numbers $$\left\{ {{\alpha _k}} \right\}_1^\infty $$ of the operator L D with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space $$W_2^\theta \left[ {0,\pi } \right],\theta > - 1$$ , we construct a function q N providing a 2N-approximation to the potential on the basis of the finite spectral data set $$\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N$$ . The main result is that, for arbitrary τ in the interval −1 ≤ τ < θ, the estimate $${\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}$$ is true, where $${\left\| {\left. \cdot \right\|} \right._\tau }$$ is the norm on the Sobolev space $$W_2^\tau $$ . The constant C depends solely on $${\left\| {\left. q \right\|} \right._\theta }$$ .

We study the cases of unique solvability of the Dirichlet problem for the Boussinesq–Love equation.

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second-order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second-order method. Copyright © 2016 John Wiley & Sons, Ltd.

We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.

We study classical solutions of boundary value problems for a nonstrictly hyperbolic third-order equation. The equation is posed in a half-strip and a quadrant of the plane of two independent variables. The Cauchy conditions are posed on the lower boundary of the domain, and the Dirichlet conditions are posed on the lateral boundaries. By using the method of characteristics, we find the analytic form of the solution of considered problems. The uniqueness of the solutions is proved.

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to $+\infty$. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.

We study the qualitative properties of the unique global viscosity solution of the superquadratic diffusive Hamilton-Jacobi equation with (generalized) homogeneous Dirichlet conditions. We are interested in the phenomena of gradient blow-up (GBU), loss of boundary conditions (LBC), recovery of boundary conditions and eventual regularization, and in their mutual connections. In any space dimension, we establish the sharp minimal rate of GBU. Only partial results were previously known except in one space dimension. We also obtain the corresponding minimal regularization rate. In one space dimension, under suitable conditions on the initial data, we give a quite detailed description of the behavior of solutions for all $t>0$. In particular, we show that nonminimal GBU solutions immediately lose the boundary conditions after the blow-up time and are immediately regularized after recovering the boundary data. Moreover, both GBU and regularization occur with the minimal rates, while loss and recovery of boundary data occur with linear rate. We describe further the intermediate singular life of those solutions in the time interval between GBU and regularization. We also study minimal GBU solutions, for which GBU occurs {\it without} LBC: those solutions are immediately regularized, but their GBU and regularization rates are more singular. Most of our one-dimensional results crucially depend on zero-number arguments, which do not seem to have been used so far in the context of viscosity solutions of Hamilton-Jacobi equations., 43 pages, 1 figure

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field., Journal of the European Mathematical Society, European Mathematical Society, In press

Bounds are obtained for the $$L^p$$ norm of the torsion function $$v_{\varOmega }$$, i.e. the solution of $$-\varDelta v=1,\, v\in H_0^1(\varOmega ),$$ in terms of the Lebesgue measure of $$\varOmega $$ and the principal eigenvalue $$\lambda _1(\varOmega )$$ of the Dirichlet Laplacian acting in $$L^2(\varOmega )$$. We show that these bounds are sharp for $$1\le p\le 2$$.

In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Let Y be the space of all entire functions f : ℂ2 → E defined by the vector valued Dirichlet series, where E is a complex Banach algebra with the unit element. We study various topologies defined on the space Y and characterize continuous linear transformations on Y.

Conditions guaranteeing well-posedness of the problem u ' ' = p 0 ( t ) u + q 0 ( t ) ${u^{\prime \prime }=p_0(t)u+q_0(t)}$ , u ( a ) = 0 ${u(a)=0}$ , u ( b ) = 0 ${u(b)=0}$ , are established. Here p 0 , q 0 : ] a , b [ → ℝ ${p_0,q_0\colon ]a,b[\rightarrow \mathbb {R}}$ are locally Lebesgue integrable functions and may have singularities at t = a ${t=a}$ and t = b ${t=b}$ .

This article investigates the growth of multiple Dirichlet series. The order and the type of n-tuple Dirichlet series in ℂ n are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained, which generalize some results about simple Dirichlet series of Lindelof and Pringsheim.

The Dirichlet problem for the heat equation in a bounded domain $$ \mathcal{G} $$ ⊂ ℝ n+1 is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.

In this paper, we investigate the existence, uniqueness and stability of pseudo almost periodic mild solution to nonautonomous impulsive integro-differential equations in Banach space. The working tools are based on the fixed point theorems and Gronwall–Bellman inequality. To illustrate our main results, we study pseudo almost periodic solution of the heat equations with Dirichlet conditions.

In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of solutions of MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and using Minty's method, we show the existence of weak solutions to the original MFG., Comment: 15 pages

3rd International Conference on Analysis and Applied Mathematics (ICAAM) -- SEP 07-10, 2016 -- Almaty, KAZAKHSTAN In this paper, we apply finite difference method to Bitsadze-Samarskii type overdetermined elliptic problem with Dirichlet conditions. Stability, coercive stability inequalities for solution of the first and second order of accuracy difference schemes (ADSs) are proved. Then, established abstract results are applied to get stable difference schemes for Bitsadze-Samarskii type overdetermined elliptic multidimensional differential problems with multipoint nonlocal boundary conditions. Finally, numerical results with explanation on the realization in two dimensional and three dimensional cases are presented. Inst Math & Math Modeling, Al Farabi Kazakh Natl Univ, L N Gumilyov Eurasian Natl Univ WOS:000439421100015 2-s2.0-85048020715

By the method of Knopp-Kojima, the generalized order of Dirichlet series is studied and some interesting relations on the maximum modulus, the maximum term and the coefficients of entire function defined by Dirichlet series of slow growth are obtained, which briefly extends some results of paper [1].

In the first part, we investigate the singular BVP $$\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u$$, u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $$\mathcal{H}$$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems $$\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)$$, u(0) = A, u(1) = B, $$\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0$$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.

We provide a new result on the existence of extremal solutions for second–order Dirichlet problems with deviation argument. As a novelty in this work, the nonlinearity need not be continuous or monotone. In order to obtain this new result, we use a generalized monotone method coupled with lower and upper solutions. Primary classification number: 34B99.

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive bounded continuous solutions with a precise global behavior for the semilinear elliptic system Δu = p(x)u α ν r in domains D of ℝn, n ≥ 3, with compact boundary (bounded or unbounded) subject to some Dirichlet conditions, where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and the potentials p, q are nonnegative and belong to the Kato class K(D).

For the Lame system with homogeneous Dirichlet conditions on the boundary of a two-dimensional polygonal domain containing a reentrant corner, the existence and uniqueness of an R ν-generalized solution in a special weighted vector set is proved. Preliminarily, the properties of such sets are examined and a weighted analogue of the Friedrichs inequality is established.